{
  "id": "1501.01050",
  "version": "v1",
  "published": "2015-01-06T01:02:00.000Z",
  "updated": "2015-01-06T01:02:00.000Z",
  "title": "On the ring of cooperations for 2-primary connective topological modular forms",
  "authors": [
    "Mark Behrens",
    "Kyle Ormsby",
    "Nathaniel Stapleton",
    "Vesna Stojanoska"
  ],
  "comment": "82 pages, 19 figures",
  "categories": [
    "math.AT"
  ],
  "abstract": "We analyze the ring tmf_*tmf of cooperations for the connective spectrum of topological modular forms (at the prime 2) through a variety of perspectives: (1) the E_2-term of the Adams spectral sequence for tmf ^ tmf admits a decomposition in terms of Ext groups for bo-Brown-Gitler modules, (2) the image of tmf_*tmf in the rationalization of TMF_*TMF admits a description in terms of 2-variable modular forms, and (3) modulo v_2-torsion, tmf_*tmf injects into a certain product of copies of TMF_0(N)_*, for various values of N. We explain how these different perspectives are related, and leverage these relationships to give complete information on tmf_*tmf in low degrees. We reprove a result of Davis-Mahowald-Rezk, that a piece of tmf ^ tmf gives a connective cover of TMF_0(3), and show that another piece gives a connective cover of TMF_0(5). To help motivate our methods, we also review the existing work on bo_*bo, the ring of cooperations for (2-primary) connective K-theory, and in the process give some new perspectives on this classical subject matter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-06T01:02:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "connective topological modular forms",
      "cooperations",
      "perspectives",
      "adams spectral sequence",
      "connective cover"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 82,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150101050B"
    }
  }
}